Quasitriangular Hopf algebra

In mathematics, a Hopf algebra, H, is quasitriangular if there exists an invertible element, R, of [math]\displaystyle{ H \otimes H }[/math] such that

• [math]\displaystyle{ R \ \Delta(x)R^{-1} = (T \circ \Delta)(x) }[/math] for all [math]\displaystyle{ x \in H }[/math], where [math]\displaystyle{ \Delta }[/math] is the coproduct on H, and the linear map [math]\displaystyle{ T : H \otimes H \to H \otimes H }[/math] is given by [math]\displaystyle{ T(x \otimes y) = y \otimes x }[/math],

• [math]\displaystyle{ (\Delta \otimes 1)(R) = R_{13} \ R_{23} }[/math],

• [math]\displaystyle{ (1 \otimes \Delta)(R) = R_{13} \ R_{12} }[/math],

where [math]\displaystyle{ R_{12} = \phi_{12}(R) }[/math], [math]\displaystyle{ R_{13} = \phi_{13}(R) }[/math], and [math]\displaystyle{ R_{23} = \phi_{23}(R) }[/math], where [math]\displaystyle{ \phi_{12} : H \otimes H \to H \otimes H \otimes H }[/math], [math]\displaystyle{ \phi_{13} : H \otimes H \to H \otimes H \otimes H }[/math], and [math]\displaystyle{ \phi_{23} : H \otimes H \to H \otimes H \otimes H }[/math], are algebra morphisms determined by

[math]\displaystyle{ \phi_{12}(a \otimes b) = a \otimes b \otimes 1, }[/math]

[math]\displaystyle{ \phi_{13}(a \otimes b) = a \otimes 1 \otimes b, }[/math]

[math]\displaystyle{ \phi_{23}(a \otimes b) = 1 \otimes a \otimes b. }[/math]

R is called the R-matrix.

As a consequence of the properties of quasitriangularity, the R-matrix, R, is a solution of the Yang–Baxter equation (and so a module V of H can be used to determine quasi-invariants of braids, knots and links). Also as a consequence of the properties of quasitriangularity, [math]\displaystyle{ (\epsilon \otimes 1) R = (1 \otimes \epsilon) R = 1 \in H }[/math]; moreover [math]\displaystyle{ R^{-1} = (S \otimes 1)(R) }[/math], [math]\displaystyle{ R = (1 \otimes S)(R^{-1}) }[/math], and [math]\displaystyle{ (S \otimes S)(R) = R }[/math]. One may further show that the antipode S must be a linear isomorphism, and thus S² is an automorphism. In fact, S² is given by conjugating by an invertible element: [math]\displaystyle{ S^2(x)= u x u^{-1} }[/math] where [math]\displaystyle{ u := m (S \otimes 1)R^{21} }[/math] (cf. Ribbon Hopf algebras).

It is possible to construct a quasitriangular Hopf algebra from a Hopf algebra and its dual, using the Drinfeld quantum double construction.

If the Hopf algebra H is quasitriangular, then the category of modules over H is braided with braiding

[math]\displaystyle{ c_{U,V}(u\otimes v) = T \left( R \cdot (u \otimes v )\right) = T \left( R_1 u \otimes R_2 v\right) }[/math].

Twisting

The property of being a quasi-triangular Hopf algebra is preserved by twisting via an invertible element [math]\displaystyle{ F = \sum_i f^i \otimes f_i \in \mathcal{A \otimes A} }[/math] such that [math]\displaystyle{ (\varepsilon \otimes id )F = (id \otimes \varepsilon)F = 1 }[/math] and satisfying the cocycle condition

[math]\displaystyle{ (F \otimes 1) \cdot (\Delta \otimes id)( F) = (1 \otimes F) \cdot (id \otimes \Delta)( F) }[/math]

Furthermore, [math]\displaystyle{ u = \sum_i f^i S(f_i) }[/math] is invertible and the twisted antipode is given by [math]\displaystyle{ S'(a) = u S(a)u^{-1} }[/math], with the twisted comultiplication, R-matrix and co-unit change according to those defined for the quasi-triangular quasi-Hopf algebra. Such a twist is known as an admissible (or Drinfeld) twist.

See also

• Quasi-triangular quasi-Hopf algebra
• Ribbon Hopf algebra

References

• Montgomery, Susan (1993). Hopf algebras and their actions on rings. Regional Conference Series in Mathematics. 82. Providence, RI: American Mathematical Society. ISBN 0-8218-0738-2. 
• Montgomery, Susan; Schneider, Hans-Jürgen (2002). New directions in Hopf algebras. Mathematical Sciences Research Institute Publications. 43. Cambridge University Press. ISBN 978-0-521-81512-3.